
We extend the single-line Pareto fronts to line-based Pareto fronts with more than one lines in three dimensions, and reveal that a uniform solution set cannot always be obtained by maximizing the hypervolume indicator. The solution set is uniform only if the hypervolume can be decomposed into independent parts.

In addition to the line-based Pareto fronts, we also investigate the plane-based Pareto fronts in three dimensions. The main result is that a uniform solution set (i.e., a solution set generated by the DAS method [1]}) is not always optimal for hypervolume maximization. It is locally optimal with respect to a \((\mu +1)\)  selection scheme.

